Multiple choice

The number of ways in which the letters of the word "PROPORTION" is arranged without changing the relative positions of the vowels and consonants is

  1. $720$
  2. $\dfrac{4!}{2!}\cdot 6!$
  3. $\dfrac{4!}{3!}\cdot 6!$
  4. $\dfrac{4! 6!}{2! 3!}$
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A Correct answer
Explanation

PROPORTION has 10 letters: P, R, O, P, O, R, T, I, O, N. Vowels are O, O, I, O (4). Consonants are P, R, P, R, T, N (6). The vowels occupy 4 fixed positions and consonants occupy 6 fixed positions. Ways to arrange vowels = 4! / 3! = 4. Ways to arrange consonants = 6! / (2! * 2!) = 180. Total ways = 4 * 180 = 720.

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